Multiply the following complex numbers, marked as blue dots on the graph: $[5(\cos(\frac{1}{3}\pi) + i \sin(\frac{1}{3}\pi))] \cdot [\cos(\frac{3}{4}\pi) + i \sin(\frac{3}{4}\pi)]$ (Your current answer will be plotted in orange.)
Answer: Multiplying complex numbers in polar forms can be done by multiplying the lengths and adding the angles. The first number ( $5(\cos(\frac{1}{3}\pi) + i \sin(\frac{1}{3}\pi))$ ) has angle $\frac{1}{3}\pi$ and radius $5$ The second number ( $\cos(\frac{3}{4}\pi) + i \sin(\frac{3}{4}\pi)$ ) has angle $\frac{3}{4}\pi$ and radius $1$ The radius of the result will be $5 \cdot 1$ , which is $5$ The angle of the result is $\frac{1}{3}\pi + \frac{3}{4}\pi = \frac{13}{12}\pi$ The radius of the result is $5$ and the angle of the result is $\frac{13}{12}\pi$.